It is supposed to rain today. (It has been forecasted to rain.)

I should wear my rain jacket because it is supposed to rain. (It would be a good idea to protect myself from the rain that I know is forecasted.)

I should go to the doctor because I have had a sore throat for over a week. (It would be a good idea.)

I was supposed to go to the doctor last week but i cancelled my appointment. (It was arranged.)

I'll add a few examples:

I'm supposed to go on a diet! (the doctor told me to)

I really should go on a diet (I think so, my friends think so, and the doctor told me to)

The President is supposed to go the China on a visit (it's been arranged)

The President should go to China (I think it would be good for international relations)

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Should is an obligation you put on yourself. Supposed to is often an external obligation. It is not as strong as must/have to, because it implies more possibility that you may decide not to do it. =)

Which What is the difference between "should" and "be supposed to"